Return to the list of my pages written in English about the two envelopes problem

Theory of "E = (1/2)2a + (1/2)a" on The Two Envelope Paradox
– Had the illusion of objective expectation made them advocate such a theory? –

On February 7, 2016 ,this page was totally revised and the title of this page was changed.

Introduction

• A brief wording of the two envelope paradox might be as follows.
  (This item was added on March 3, 2016)
  • Each of two envelopes contains money.
  • One envelope contains twice as much money as the other. In other words, if the lesser amount = a then the greater amount = 2a.
  • Randomly you choose one envelope.
  • Let x be the amount of money in the chosen envelope and let e be the expectation of the amount of money in the other envelope. (← Revised on March 6, 2016)
    Then e = (probability 1/2) (x/2) + (probability 1/2) 2x = 1.25x > x.
  • Because this calculation does not depend on amount x, the other envelope is always profitable than the chosen envelope.
  • It contradicts the symmetry of these envelopes. Paradox!!
• Some people think as follows.
  Following two expectation formulas are very similar.

  E = (1/2)2x + (1/2)(x/2).
  E = (1/2)2a + (1/2)a.

  The former formula is a mistake in writing of the latter formula , because the latter formula causes no paradox.By this finding, the two envelope paradox was resolved..
  I call this opinion Theory of "E = (1/2)2a + (1/2)a". 
  And I call the advocators of this opinion DivideThreeByTwoians, because (1/2)2a + (1/2)a = (3/2)a.
  ↑ Added on Novenber 3, 2015.
   
• Some of these people have made fictions to rationalize their opinions.
  One of these fictions is like this.
  In the expectation formula E = (1/2)2x + (1/2)(x/2) following values are different.
  If the amounts of money in the two envelopes are A and 2A, then

  • The value of x in the term (1/2)2x is A.
  • The value of x in the term (1/2)(x/2) is 2A.

  This is the cause of the paradox.
  ↑ Revised on July 5, 2015.
  I call this opinion "Inconsistent variable theory" or IVT.
   
• Inconsistent variable theory is minor among mathematicians,  but some articles in some Wikipedias introduced it as one of the main opinions.
  • Article "The two envelopes problem" (revision at 03:02, 2 May 2015) in the Engilish language Wikipedia.
  • Article "Paradoxe des deux enveloppes" (en date du 28 septembre 2014 à 20:48) in the French language Wikipedia.
   
• To my surprise, there were a mathematician who advocated the inconsistent variable theory.
  And more surprisingly there were a psychologist who did not think that a fallacy of probability is the cause of the paradox.
  These fact made me wonder as follows.
  Had there been somebody who used variable symbol inconsistently and felt a paradox on the two envelopes problem?
  (This row was added on November 29, 2015, and revised on December 27, 2015.)
   
• I have studied the "Inconsistent variable theory".and found that there are relatives of it.
  (This row was revised on September 23, 2015, February 7, 2016.)
  
  One of these relatives is as follows.
  If we think only one pair of amounts of money (A, 2A) we can avoid the paradox.
  Therefore the thinking of two pairs of amounts of money (x/2, x) and (x, 2x) are the cause of the paradox.
  I call this opinion "Not three amounts theory".
  It seems that this opinion and the Smullyan's paradox on the two envelopes problem are result of same fallacy.
  
  
• "Inconsistent variable theory" and "Not three amount theory" are logically similar but are psychologically contradictory.
  It suggests that both opinions are sophism and that the advocators of these theories have resolved a fictitious paradox.
  (I wrote the companion page "The relatives of IVT on the two envelopes problem" about these fictions and some other fictions.)
  
  
• I think that the following phenomena have not arose in their brain, because none of them had reported that they had experienced these phenomena.
  • Inconsistent use of variable symbol
  • Thinking three amounts of money against their will
  ↑ This item was added on December 13, 2015.
  
  
• But I did not yet know which of next is right.
  • When they felt a paradox on the two envelopes problem, two pairs of amount of money existed in their brain.
    And to resolve this paradox they changed the problem.
  • They could not feel any paradox themselves but they made a fiction about the mind of the people who could feel a paradox.
  • They thought that it is the resolution of the paradox to find a correct calculation formula ← Added on February 7, 2016.
  And I did not yet know which of next is their aim.
  • They want to avoid mathematical thinking.
    • They want to avoid to think of two pairs of amounts of money.
    • They want that the probabilty is always 1/2.
    • They want to avoid to think the probability 1/2 as a conditional probability.
  • They simply want to find a way that convince them of the equivalence of the two envelopes.
    ↑ This item was added on December 5, 2015.
  • They were engaged in jobs related to theory of probability, so it was necessary for them to pretend to be able to solve the problem.
    ↑ This item was added on December 12, 2015.
• In any case、they looked like sorcerers to my eyes.
  
  ↑ This drawing was added on July 4, 2015.
  
  
• But on April 4, 2016, I conceived new idea which is a synthesis of two ideas about illusions of expetation.
  
  DivideThreeByTwoinas had been caught by illusion of objective expectation.
  In other words, in their mind the expected value is conservative quantity like energy or mass.
  
  On the other hand they had been caught by illusion of objective equivalence.
  In other words, in their mind the equivalence of the two envelopes is not varying, regardless of the amount of money in the chosen envelope.
  
  Because DivideThreeByTwoians had been caught by both illusions, they conceived the theory of  "E = (1/2)2a + (1/2)a" along with the following flow of thought.
  
  

  Illusion of objective equivalence let them feel an unusual paradox when they read following calculation.

  Let x be the amount of money in your enveloe.
  Then the expected value of the other envelope is (1/2)2x + (1/2)(x/2) > x.

  This moment usual people did not yet feel a paradox and usual people felt usual paradox at the moment when they noticed following thought.

  Because this calculation does not depend on the value of x, the other envelope is always more profitable than the chosen envelope.

  ↓

  Illusion of objective equivalence let them seek a expectation formula which calculate same value for each of the two envelopes.

  ↓

  They found that expectation formula "E = (1/2)2a + (1/2)a" meets this criteria.

  ↓

  Illusion of objective expectation let them mistake this formula for the one and only correct formula.

  
  About the paradox they felt, I call it "Fictitious paradox by illusion of objective equivalence and illusion of objective expectation".
  ↑ Revised on April 5, 2016.
  
    ↑ Added on April 10, 2016.
  
  
    ↑ Added on April 15, 2016.
   
• Birds of a feather
  
  Some people think that the equivalence of the two envelopes are kept even after opening one envelope.
  I think that these people had been caught by illusion of objective equivalence.
  
  Example
  One of them wrote R script to prove his opinion. Because he was good writer of R script, his script calculated extremely accurate expectation and let him down.
  
  In my perception, illusion of objective expectation and illusion of objective equivalence are birds of a feather.
  ↑ Added on April 5, 2016.

The process through which the Two Envelope Paradox arise

The process through which the Two Envelope Paradox (Exchange Paradox) arise is as follows.

• There are two envelopes.
• One envelope contains twice as much money as the other.
• Randomly you choose one envelope.
• Let A be the chosen envelope and let B is the another envelope.
  Then equally likely, B contains twice as much money as A, or B contains half money of A.
• Let e be the expectation of the amount of money in the envelope B while the amount of money in A is x. Then
  e = (probability 1/2) (x/2) + (probability 1/2) 2x = 1.25x. · · · (1)
• It means that e is greater than x, and you should trade envelope A with B.
• Because this calculation does not depend on x, the other envelope is always more profitable than the chosen envelope.
• But there is no reason of a gap between two envelopes.
• It is a paradox !

↑ It was revised on February 8, 2015. March 3, 2016.

Mathmatical resolution of the Two Envelope Paradox

The mechanism of this paradox is disappointingly simple.

We forget the odds of the pair (x/2 , x) , and the odds of the pair (x , 2x).
That is why we thoughtlessly assign 1/2 to the probabilities of the events which appear in the calculating formula of the expected amount of money. (This sentence was revised on February 27, 2016.)

This is a kind of probability illusion called "Base Rate Fallacy", and it is the cause of the two envelope paradox.


(This figure was added on September 20, 2014, and was moved here on March 14, 2015.)

Following figure is an explanation of correct calculation of probability.


(This figure from a companion page "An outline of the 'Two envelopes problem'" was added on February 28, 2016.)

Note:
The equation (1) should be corrected like this.

e = (some probability p) (x/2) + (1 − p) 2x = (some rate r)x

Even if the opposite envelope is more favorable for a value of the amount of money in the chosen envelope, there is no wonder.
And if the opposite envelope is favorable for a value of the amount of money in the chosen envelope, the opposite envelope must be unfavorable for some value of the amount of money in the chosen envelope.
So the equivalence of the two envelopes are kept.
↑ These sentences were added on April 5, 2016, and were revised on April 6, 2016.

There remains no paradox.
Only mathematical problems remain. ( → Some example)

Using conditional expectation, some mathematicians proved the equivalence of the two envelopes.
And I also did it on February 2016.
Please see a companion page "Two methods for the proof of the equivalence of the envelopes of the two envelopes problem".

Inconsistent variable theory for the Two Envelope Paradox

To my suprise some people don't think so.

They explain the cause of the paradox as follows.


In the following sections,  I call such a opinion 'Inconsistent variable theory' or IVT.


(This figure was added on September 20, 2014, and revised on May 4,2015)

Not three amounts theory for the Two Envelope Paradox

I found that some other people advocate similar but different opinion.

They explain the cause of the paradox as follows.


Or they explain the cause of the paradox as follows.


In the following sections,  I call such a opinion  'Not three smounts theory'.

Mystery of the theory of "E = (1/2)2a + (1/2)a" is more mysterious than the two envelope paradox

This section was added on January 10, 2016.

I have been interested about the mind of advocators of the theory of "E = (1/2)2a + (1/2)a".


The advocators of the theory of "E = (1/2)2a + (1/2)a" are very minor, and I have not found articles which discusses about the mind of them.  But I cannot stop thinking about their mind.

The two interpretations of the problem

Usually the problem statement is interpreted like this.



But it appears that the followers of IVT interpret the problem like this.

Corresponding Mental Models

The former interpretation corresponds with this mental model.



In the following sections , the people with this mental model are called 'DoublePairian'.

---

The latter interpretation corresponds with this mental model.



In the following sections , the people with this mental model are called 'SinglePairian'.

(Note : Some theory says that people may create another mental model which is similar to the SinglePairian's mental model".  Such people should be called "LesserOrGreaterian".)

Two 'Two Envelope Problems'

'The Two Envelopes problem' for the SinglePairians and 'The Two Envelopes Problem' for the DoublePairians differ widely from each other.
Using mathematical notation, we can demonstrate the difference of the two problems.

---

Let x and y be the amounts in the envelope selected by you and the amount in the another envelope respectively.
Let X and Y be random variables which take x and y as their value respectively.
Let P be a random variable which takes the pair of amounts in the envelopes as it's value.
Let a be the lesser amounts in the two envelopes.

	the DoublePairian's problem (mathematicians prefer this)	the SinglePairian's problem (philosophers prefer this)
the condition on which the expectation is calculated	the amount of the envelope first selected by you	the pair of the amount
pairs of the amount	two pairs (x/2, x) and (x, 2x) x is the amount of the selected envelope	one pair (a, 2a)
what are compared in the Closed version Problem	random variable E(Y|X) vs random variable X	conditional expectation E(X|P=(a, 2a)) vs conditional expectation E(Y|P=(a, 2a)) ↑ revised on July 20, 2015
what are compared in the Opened version Problem	conditional expectation E(Y|X=x) vs value x ↓ considering any x ↓ random variable E(Y|X) vs random variable X	nothing
paradox	E(Y|X=x) = (1/2)2x + (1/2)x/2. ∴ for any x E(Y|X=x) > x. In other words, E(Y|X) > X. If E(X) and E(Y) are finite, E(Y) > E(X) . under symmetry E(X) > E(Y) . ∴ E(X) > E(X) and E(Y) > E(Y) . ↑ revised on January 24, 2015	Let p be the event P = (a, 2a). E(Y|p)= (1/2)E(X/2|p ∧ X=2a) + (1/2)E(2X|p ∧ X=a). ∴ E(Y|p) = (1/2)E(X/2|p) + (1/2)E(2X|p) = 1.25E(X|p) > E(X|p). If E(X) and E(Y) are finite, E(Y) > E(X) . under symmetry E(X) > E(Y) . ∴ E(X) > E(X) and E(Y) > E(Y) . ↑ revised on March 30, 2015
invariant	the expectation formula must contain term of x/2 , and term of 2x	the probabilities of the terms in the expectation formula must be 1/2

↑ This description was added on November 5, 2014, and revised on May 10, 2015.

Mystery of mind of advocators of the theory of "E = (1/2)2a + (1/2)a"

The "Inconsistent variable theory" and the "Not three amounts theory" is psychologically doubtful and accompanied by some illogical aspects.
(↑ Revised on December 30, 2015)

Indeed these theories induce many questions.

On July 7, 2015, I classified questions as follows.

These theories are unbecoming to "PARADOX".

This paragraph was revised on January 10, 2016.

• Did they think that the famous mathematicians who created the problem had used inconsistent variable?
  Or did they think that the mathematicians had thought of three amounts of money against own will?
   
• Did they think that many mathematicians who had read and spread the problem had used inconsistent variable?
  Or did they think that the mathematicians had thought of three amounts of money against own will?
   
• Did they think that such a elementary mistake in writing calculation formula of expectation had created a famous paradox?

Psychological phenomenon of inconsistent variable symbol is not realistic.

• The process to a paradox with inconsistent variable is as follows.
  • Denote by x the amount in the chosen envelope,
    and denote by y the amount in the another envelope,
    and denote by e the expected value of y.
  • y = x/2 with a probability 1/2, and y = 2x with same probability.
  • e = (1/2)(x/2) + (1/2)2x. · · ·  (1)
  • e = (1/4)x + (1/4)4x. · · ·  (2)
  • e = (5/4)x. · · ·  (3)
  • e > x. · · ·  (4)
  • Smell a paradox. · · ·  (5)
  • Check the expectation formula.· · ·  (6)
  • Become sure of the existence of a paradox. · · ·  (7)
  To reach the step (5), I need four steps from (1) through (4).
  At the step (6), I take several minutes.
  
  Had there been somebody who did not notice before the step (7) that the symbol x has different values?
  (This question was added on December 22, 2014, and revised on May 09, 2015, December 30, 2015.)
   
• We can find the same paradox in the opened version two envelopes problem.
  (In the opened version problem, the chance to trade the envelopes is given after you open your envelope)
  But in the opened version problem, nobody can use inconsistent variable.
  
  Can we switch cognitive mechanism depending on the version of the problem?
   
• In our brain, we can use mental variable symbol inconsistently.
  Especially if the problem does not refer to calculation formula of the expected value, it is easy.
  
  But seeing written variable symbol, it is very hard to use variable symbol inconsistently.
  Inconsistent variable symbol in mind
  
  (This item was added on November, 2015)

There are some Wikipedia articles which do not introduce such theories.

• To my eyes, following Wikipedia articles seem not to contain the issue of the "closed version" problem and the theory of "E = (1/2)2a + (1/2)a".
  "Umtauschparadoxon" (am 14. Juni 2014 um 08:43) in the German language Wikipedia
  '"Задача о двух конвертах" (14:19, 10 июля 2014) in the russian language Wikipedia
  (In the colosed version problem, before you open your envelope, the opportunity of trading envelopes are given.)
  
  If we easily use the inconsistent variable symbol or easily think of three amounts of money agaist our will, why these articles do not introduce closed version problem nor the theory of "E = (1/2)2a + (1/2)a" ?
  (This question was added on May 17, 2015, and was revised on Feruary 7, 2016.)

There are other opinions that are different but similar.

• I found a article which was as follows.
  In the beginning, it was described that the amount in the envelope which you chose first is smaller amount A or larger amount 2A.
  The expectation formula which was presented as wrong version contains the mean value of the smaller amounts and the mean value of larger amounts.
  The author had no doubt about the probability 1/2.
  It was described that the cause of paradox is a wrong assumption that amount and its side (smaller side or larger side) are independent, and this wrong assumption mistakenly induces that the mean value of the smaller amounts and the mean value of larger amounts are same.
  I wonder why such a author had not reached IVT nor the "Not three amounts theory" and had reached another reasoning.
  (This question was added on September 13, 2014.)

Advocators of the "Inconsistent variable theory" seem to have not felt any paradox themselves.

• Many mathematicians tried to explain why people made a fallacy of probability.
  Example: Wrong assumption that always the odds of each pair of amounts of money are equal is the cause of the paradox.
  But I have never found an advocator of the "Inconsistent variable theory" who tried to explain why people used inconsistent variable symbol.
  
  Did they think it real phenomenon?
  (This question was added on May 30, 2015)
   
• If one have used inconsistent variable symbol, he/she must claim the IVT as a theory about true phenomenon.
  But some people claim the IVT as a hypothesis.
  
  Have they really used inconsistent variable symbol themselves?
  (This question was added on November 11, 2014)

Advocators of the not three amounts theory seemed have felt standard paradox (DoublePairian's paradox) and have resolve SinglePairian's paradox..

• Advocators of the "Not three amounts theory" often pointed out several mistakes of standard resolution (Mathematial rsolution, DoublePairian's resolution).
  This suggest that they were bothered by thinking three amounts of money.
  
  Were they DoublePairian when they felt a paradox?

These theories look like sophism.

• "Inconsistent variable theory" has following illogical aspect.
  No evidence of such a phenomenon has been presented.
  "Not three amounts theory" has following illogical aspect.
  It has not been proven that we can not think of three amounts of money with no paradox. (← Revised on March 18, 2016)
  Why could they believe their opinions?
  (This question was added on December 27, 2015.)
   
• Advocators of IVT not seldom describe the "Not three amounts theory" at the same time.
  But these opinions are in conflict with each other.
  
  Had they really experienced these phenomena?
  (This question was added on May 30, 2015, and revised on October 10,2015, December 27, 2015.)
   
• Advocators of these opinions lay emphasis on how their analysis prove the equivalency of the two envelopes.
  
  Are their analyses based on real psychological phenomena?
  (This question was added on November 10, 2014.)
   
• In my perception, the advocators of these opinions strongly condemn the opinion that a fallacy of probability is the cause of the paradox.
  
  If they really have experienced the phenomenon of inconsistent variable symbol, why they deny another phenomenon that other people have experienced?
  (This question was added on May 10, 2015. And it was revised on May 31, 2015.)
  
  If they really dislike to think of three amounts of money, why they deny the fact usual person like to think of three amounts of money?
  (This question was added on February 7, 2016.)

Who wanted to spread such a paradox?

On July 20, 2015, this question was added.

• If some person used inconsistent variable symbol, and felt some mystery, and spread his experience as a paradox, he simultaneously spread his less ability of mathematics.
  Who did aim to do so?

Why did they pretend not to see the standard problem and paradox?

On July 10, 2015, this paragraph was added and revised on February 7, 2016.

• In my perception most articles by the advocators of the theory of "E = (1/2)2a + (1/2)a" seemed not to refer to the articles which discussed standard problem (DoublePairian's problem) and standard resolution (DoublePairian's resolution) with a few exceptions.
  Why they did not so?

Some hypotheses about the mind of the SinglePairians who conceive of the theory of "E = (1/2)2a + (1/2)a"

This was placed on February 9, 2015, as a refinement of paragraph "Some hypotheses about the proccess that SinglePairians conceive of the "Inconsistent variable theory" in their mind".
On May 23, 2015, the structure of this paragraph was changed.

Hypotheses about the paradox which they themselves felt

My main hypothesis

• They felt a fictitious paradox by illusion of objective equivalence and illusion of objective expectation.
  ↑ Rrevised on May 15, 2016.
  
  They had been caught by illusion of objective expectation.
  In other words, in their mind the expected value is conservative quantity like energy or mass.
  On the other hand they had been caught by illusion of objective equivalence.
  In other words, in their mind the equivalence of the two envelopes is not varying, regardless of the amount of money in the chosen envelope.
  So they could feel paradox when they read any expectation formula which probably calculates unequivalent expectation.
  They found that equation "E = (1/2)2a + (1/2)a" does not calculate unequivalent expectation.
  This made them think they had found the resolution.
  For them to find the mistake of the formula which was described in the problem was not the main theme.
  This hypothesis was added on February 7, 2016, and revised on April 10, 2016, May 15, 2016.
  
  
    ↑ Added on May 15, 2016.
  

Other hypotheses

• They thought that the paradox which is to be resolved is the crisis of the equivalence of the two envelopes.
  
  They were afraid that the equivalence of the two envelopes had been endangered.So they searched a fashion of thinking which guarantees the equivalence.
  This hypothesis was added on October 12, 2015.
  
  
• They did not feel any paradox.
  A possibility.
  
  They doubted the variable symbol from the beginning.
  So they could no longer feel any paradox.
  Another possibility.
  
  They did not understand the expectation formula.
  Therefore they could not find any paradox.
  ↑ This possibility was added on October 12, 2015.
  One more possibility.
  
  Before they felt a paradox, they found the opinion that "E = (1/2)2a + (1/2)a" is the correct calculation formula.
  Their brain were unflexible so that they could only have the SinglePairan's mental model.
  So they thought that there are no other correct calculation formula.
  Therefore they could not find any paradox.
  ↑ This possibility was added on December 12, 2015.
  
  
• They had felt the standard paradox (DoublePairian's paradox) and had resolved another paradox.
  
  When they read the expectation formula, they got the DoublePairian's mental model and felt the DoublePairian's paradox.
  The paradox which they resolved was not the paradox which they felt. (This sentence was added on December 31, 2015.)
  
  
• They really had felt the SinglePairian's paradox.
  
  They used inconsistent variable symbol themselves.
  And they felt a paradox without noticing this mistake.

Hypotheses about the mental proccess of understanding the expectation formula

This paragraph was added on September 26, 2015, and revised on May 15, 2016.

My main hypothesis

• They first understood variable symbol. It is the usual process of understanding the expectation formula in the two envelopes problem.
  
  Process of usual understanding.

  • Read the expectation formula. E = (1/2)2x + (1/2)(x/2).
  • Understand varible symbol. E = (1/2)2x + (1/2)(x/2).
  • Understand probability. E = (1/2)2x + (1/2)(x/2).
  This hypothesis was added on May 15, 2016.

Other hypotheses

• They first understood probability, therefore they could not doubt probability.
  
  Process of unusual understanding.

  • Read the expectation formula. E = (1/2)2x + (1/2)(x/2).
  • Understand probability. E = (1/2)2x + (1/2)(x/2).
  • Understand variable symbol. E = (1/2)2x + (1/2)(x/2).
  If this hypothesis is true many of DivideThreeByTwoinas would have advocated the inconsistent variable theory (IVT).
  But DivideThreeByTwoians who advocated IVT are not so many.
  Therefore this hypothesis might be untrue.
  This hypothesis was revised on May 15, 2016.
  
  
• They had no need to understand the expectation formula.
  
  For them any expectation formula which calculate different value for each envelopes was wrong.
  So they had no need to understand it.
  This hypothesis was added on April 10, 2016.
  
  
• They conceived the SinglePairian's expectation formula before they read the expetation formula in the two envelopes problem.
  
  When their eyes caught the phrase "expectation" in the two envelopes problem, they started to create calculation formula by own hands.
  And they conceived the SinglePairian's expectation formula "E = (1/2)2a + (1/2)a" before they had read the expetation formula in the two envelopes problem.
  This hypothesis was added on December 30, 2015.

Hypotheses about the trigger by which they get the SinglePairan's mental model

My main hypothesis

• They pretended SinglePairians to advocate the theory of "E=(1/2)2a + (1/2)a".
  
  They changed their mental model to the one which fit to the theory.
  This hypothesis was added on February 13, 2016.

Other hypotheses

• They understand that the two envelopes problem start at the setting of money.
  
  They make much of the fact that the pair of amounts of money will be fixed at the setting of money.
  Thereafter they can not think of two pairs of amounts of money.
  
  
• They simply confuse the number of envelopes with the number of possible amounts of money.
  
  They notice that the number of amounts of money that the expectation formula assumes is three ( x, x/2 and 2x ),  and they remember that the number of envelopes is only two.
  They think that the coexistence of x/2 and 2x in the expectation formula is a mistake,  and they jump to conclusion that the expectation formula must contain only two amounts A and 2A.
  
  
• They were influenced by suggestion.
  A possibility.
  
  When they feel a paradox they are DoublePairian.
  "Inconsistent variable theory" influence them as a suggestion.
  They forget that they have been DoublePairian and think that they have used an inconsistent variable symbol.
  This possibility was added on February 11, 2015.
  Another possibility.
  
  They were simply influenced by the article "The two envelopes problem" in the English language Wikipedia.
  From 2005, without a break that article had treated the theory of "E = (1/2)2a + (1/2)a" as common or simple resolution.
  This possibility was added on March 7, 2016.
  
  
• SinglePairians are so inflexible that they cannot switch mental model.
  
  When they read the rule of the game they got the SinglePairian's mental model.
  Thereafter, when they read the expectation formula they could not switch mental model to the DoublePairian's mental model.
  
  
• SinglePairians are so good at mathematics that they interpreted the expression "the other envelope contains X/2" as a conditional expression.
  
  They read following expressions.
  "The other envelope contains 2X with a probability 1/2. And it contains X/2 with a probability 1/2."
  They interpret these expressions as conditional expressions about the relation between amounts of money in the two envelopes,  and they do not interpret these expressions as calculation formulas.
  
  
  Then they strengthen the SinglePairian's mental model in their mind, and assign different values to the same symbol X case by case.

Hypotheses about whether they themselves used an inconsistent variable symbol or not

My main hypothesis

• They used consistent variable symbol because they were DoublePairians when they felt a paradox.
  
  There were two pairs of amount of money in their mind when they felt a paradox.
  Therefore their variable symbol was consistent.
  This hypothesis was added on May 30, 2015.

Other hypotheses

• They didn't because they had no need to understand the expectation formula.
  
  For them any expectation formula which calculate different value for each envelopes was wrong.
  So they had no need to understand it.
  This hypothesis was added on April 10, 2016.
  
  
• They used inconsistent variable symbol themselves, and they felt a paradox themselves.
  
  They assigned different value to same symbol in the different terms, and they did not notice that they had done it.
  When they read that a paradox is induced by the expectation formula, they felt a paradox in their mind.
  
  
• They didn't, because they noticed the inconsistency of the variable symbol.
  
  They did not understand the expectation formula because they have SinglePairian's mental model.
  Without understanding the formula, they noticed that the values of same variable symbol of the different terms are different.
  This hypothesis was revised on May 30, 2015, April 10, 2016.
  

Hypotheses about the trigger by which they noticed the theory of "E = (1/2)2a + (1/2)a"

My main hypothesis

• Finding that there is very similar expectation formula that causes no paradox, they mistook the paradoxical expectation formula for a mistake of it.
  
  They noticed that following two expectation formulas are very similar.

  E = (1/2)2x + (1/2)(x/2).
  E = (1/2)2a + (1/2)a.

  They thought that the former formula is a mistake in writing of the latter formula , because the latter formula causes no paradox.
  This hypothesis was added on July 4, 2015.

Other hypotheses

• They noticed their own behavior.
  
  They used an inconsistent variable symbol themselves.
  And they noticed that they had done it.
  Or they thought of three amounts of money against their will.
  And they noticed that they had done it.
  
  
• Applying the SinglePairan's mental model, they interpreted the expectation formula.
  
  They can only doubt the variable symbol x rather than the probability 1/2 in the expectation formula.
  So all that they can do is to assign different values to the same variable symbol.
  
  

Hypotheses about what they think the root cause of the paradox

My main hypothesis

• They think that wrong expectation formula is by itself the root cause.
  
  They think that wrong expectation formula is by itself the root cause.
  Therefore they rarely have interest about psychological mechanism which derives wrong expectation formula.
  This hypothesis was added on May 14, 2016.

Other hypotheses

• They have no interest about the root cause.
  
  They have not experienced the fallacies which they think as the cause of the paradox.
  Therefore they have no interest about psychological mechanism of the fallacies.
  This hypothesis was added on October 10, 2015.
  
  
• They think that the DoublePairian's mental model is the root cause.
  
  They have found that if they think only one pair of amounts of money then the paradox will vanish.
  Then they get an idea that to think two pairs of amounts is the cause of the paradox.
  They make a story of IVT to rationalize their thought.
  I think that the same fallacy as the cause of the "Smullyan's paradox on the two envelopes problem" has made them think as above.
  
  
• They think that the inonsistent use of a variable symbol is the root cause.
  
  They can not understand the expectation formula at glance.  They can only understand each terms of the expectation formula one by one.
  They assign different value to same symbol in the different terms, and they do not notice until they feel a paradox.
  
  

Hypotheses about why they want to advocate their opinions.

This paragraph was revised on October 11, 2015.

My main hypothesis

• They simply want to find a way that convince them of the equivalence of the two envelopes.
  In other words, they simply want to find a calculation formula that do not contradict the illusion of objective equivalence.
  
  They thought that the paradox which is to be resolved is the crisis of the equivalence of the two envelopes.
  Therefore they had small interest about the psychological mechanism of the fallacy which they imagined as the cause of the paradox.
  This hypothesis was added on December 5, 2015, and was revised on March 9,2016, April 10, 2016.

Other hypotheses

• They deeply believed that the standard resolution (DoublePairian's resolution) was wrong.
  This hypothesys was added on February 7, 2016.
  
  
  Following properties of them let them believe that the standard resolution (DoublePairian's resolution) was wrong.

  • They doubted the standard resolution (DoublePairian's resolution) because it did not persuade them.
  • They could not distinguish standard problem (DoublePairian's problem) and their problem (SinglePairian's problem).
  
  
• They wanted to prove that mathematical ability is not needed to resolve the two envelope paradox.
  
  They thought that the two envelopes problem has low degree of difficulty.
  But they could not understand standard resolution (DoublePairian's resolution).
  To get rid of this contradiction they made fictitious resolutions.
  
  
• They wanted to prove that their mistakes are not uncommon.
  
  Possible reasons.

  • They really used inconsistent variable symbol themselves.
  • Or they carelessly thought of three amounts of money though they disliked to do so.
  
  
• They wanted to exclude conditional expectation.
  
  Possible reasons.

  • They wanted to exclude to think of two pairs of amounts of money.
  • Or they wanted that the probability is always 1/2.
  • Or they wanted to exclude to think the probability 1/2 as a conditional probability.
  
  
• They were engaged in jobs related to theory of probability, so it was necessary for them to pretend to be able to solve the problem.
  This hypothesis was added on December 12, 2015.

I think that it is not so difficult to let DivideThreeByTwoians change their opinion.

This paragraph was added March 7, 2016 and was revised on March 15, 2016.

I think if somebody does experiment using following wording of the two envelopes problem, participants of it will feel paradox. And I think if the result of the experiment is presented to DivideThreeByTwoian, many of them will change their thought.

Wording to be used in the experiment

• Each of two envelopes contains money.
• One envelope contains twice as much money as the other. In other words, if the lesser amount = a then the greater amount = 2a.
• The expected values of amount of money in these two envelopes are both the mean value of amount a and amount 2a.
  In other words, they are both (1/2)a + (1/2)2a.
• Randomly you choose one envelope.
• Let x be the amount of money in the chosen envelope then the amount of money in the other envelope may be x/2 or 2x.
• Let e be the expected value of the amount of money in the other envelope, then e = (probability 1/2) (x/2) + (probability 1/2) 2x = 1.25x > x.
• This calculation does not depend on amount x, threfore the other envelope is always profitable than the chosen envelope.
• It contradicts the symmetry of these envelopes. Paradox!!

If the participants of this experiment feel paradox, the theory of "E = (1/2)a + (1/2)2a" can not resolve the paradox they feel.

Some hints by a Wrong explanation of the existance of a paradox.

On April 7, 2016, this paragraph was added. On May 14, 2016, this paragraph was revised.

Hint 1
"Opened virsion" problem frequentry explain the existence of the paradox with following wording.
Hypothesis 1
Hint 2
"Closed version" problem frequently explain the existence of the paradox with following wording.
Hypothesis 2
I guess that the illusion of materialized expectation is the source of the illusion of objective equivalence and the illusion of the objective expectation.
↑ Added on May 14, 2016.

Some hints by a DivideThreeByTwoian's opinion.

On February 14, 2016, this paragraph was added and was revised on February 27, 2016. On May 14, 2016, this paragraph was revised.

I had read a discussion which was made by two persons.
One is a DuublePairian and another is a DivideThreeByTwoian.
This discussion was held on late 2015 on the blog page of the former person.

After study of this DivideThreeByTwoian's opinon, I found some hints.

Hint 1
In the beginning of the discussion, the DivideThreeByTwoian presented two expectation formula.
The former was E=(1/2)2x + (1/2)(x/2) based on DoubleParirian's mental model, and the latter was E=(1/2)2a + (1/2)a based on SinglePairian's mental model

Hint 2
This DivideThreeByTwoian's opinion has big illogical aspects.

• With no reason he shifted attention to SinglePairian's mental model.
• With no resson he said that "E=(1/2)2x + (1/2)x/2" and "E=(1/2)2a + (1/2)a" are antinomy.
  It is obvious that later equation derive no paradox. Therefore if he had thought logically he would not argue it.(← Revised on March 6, 2016, March 12, 2016)
• On the pretext of the wrongness of "E=(1/2)2x + (1/2)x/2" he rejected it totally. And he did not notice the reformability of probability 1/2. (← Revised on March 6, 2016)

Hint 3
I found a phrase "your approach was subjective" in this DivideThreeByTowian's post.

Hint 4 (← Added on March 6, 2016, and revised on March 12, 2016)
To my surprise, this DivideThreeByTwoian noticed that the probability P(X is greater | X=x) and P(X is lesser | X=x) cannot always be 1/2.  
But this finding could not let him try to correct probablity 1/2 in "E= (1/2)2x + (1/2)(x/2)".

Hint 5 (← Added on March 9, 2016)
This DoublePairian's main theme was the true calculation formula of the probability.
And the second theme was proof that with this calculation formula we can prove the equivalence of the two envelopes.
But this DivideThreeByTwoian did as follows.

• He had no interest in the main theme.
• He thought that the goal of the second theme is the same as his opinion.
  In fact
  The second theme is related to the law of total expectation.
  This DivideThreeByTwoian's opinion is only related to commutative law of addition.
  So they are substantially different.

Following hypotheses are suggested by these hints.

Hypothesis 1
Hypothesis 2

Some hints which was found in an article which was written by a DivideThreeByTwoian.

On May 14, 2016, this paragraph was added.

I found a hint in a paper written by a DivideThreeByTwoian.

Hint 1
His explanation contained following opinion.


But in my eyes, there is no ambiguity.
If he is a SinglePailian the former should be random variable, and if he is a DoublePairian the latter should be random variable.
His opinion might be a sophistry.

Hint 2
His explanation contained following opinion.


But in my eyes, this opinion had following defects.

• Psychologically this opinion was in conflict with the above opinion.
• Mathematically it had no rationale.

Following hypothesis is suggested by these hints.

Hypothisis

Characteristics which are common to a substantial proportion of DivideThreeByTwoian's opinions

On March 18, 2016, this paragraph was added.

I found that a substantial proportion of DivideThreeByTwoian's opinions have same characteristics such as follows.

Characteristic 1
Characteristic 2 This characteristic was added on March 31, 2016.

Characteristic 3 This characteristic was added on April 9, 2016.

To my eyes these characteristics suggest following hypothesis.

Possible flow of thought in the mind of advocators of the theory of "E=(1/2)2a + (1/2)a"

On May 23, 2015, this paragraph was added, and revised on December 13, 2015, and on March 2, 2016.

I think that the following hypothesis is highly probable.

• When they felt a paradox on the two envelopes problem, two pairs of amount of money existed in their brain.
  And to resolve this paradox they changed the problem.

Because the following facts support this opinion.

• None of them had reported that they had experienced these phenomena.
  • Inconsistent use of variable symbol
  • Thinking three amounts of money against their will
• They often said that the mathematical solution which think of two pairs of amounts of money has not enough generality.
• They often said that the mathematical solution which think of two pairs of amounts of money are correct only before the arrangement of money.

And I think that the following hypothesis is highly probable too.

• When they read the equation they felt a paradox which was derived by the illusion of objective equivalence.
• The illusion of objective expectation let them mistake the expectation formula "E=(1/2)2a + (1/2)a" for one and only correct formula.

↑ Revised on April 10, 2016.

Because the following fact supports this opinion.

• They totally reject the equation "E = (1/2)2x + (1/2)(x/2)" and illogically shifted attention to the equation "E = (1/2)2a + (1/2) a".

According to the above I think that the following flows seem realistic.

Possible flow 1: They had been caught by the illusion of objective equivalence and the illusion of objective expectation.

I think this flow is most possible.

Questuion	Answer
Which paradox did they themselves feel?	They felt a fictitious paradox which was derived by the illusion of objective equivalence and the illusion of objective expectation.
How did they get the SinglePairan's mental model?	When they found "E = (1/2)2a + (1/2)a" they became SinglePairian.
Did they themselves use an inconsistent variable symbol?	No they did not.
What did let them notice the inconsistent use of a variable symbol?	Analysis of the expectation formula from SinglePairian's point of view,
What did they think the root cause of paradox?	DoublePairian's mental model

↑ Revised on April 10, 2016.


  ↑ Added on May 15, 2016.

I think that the following flows are not so possible.

Possible flow 2: They forgot that they were DoublPairians.

Questuion	Answer
Which paradox did they themselves feel?	DoublePairian's paradox.
How did they get the SinglePairan's mental model?	They found that there were no paradox if they have the SinglePairian's mental model.
Did they themselves use an inconsistent variable symbol?	No they did not.
What did let them notice the inconsistent use of a variable symbol?	Analysis of the expectation formula from SinglePairian's point of view.
What did they think the root cause of paradox?	DoublePairian's mental model

But I cannot throw away the following flow., too.

Possible flow 3: They made a fiction about the mind of the people who felt a paradox.

Questuion	Answer
Which paradox did they themselves feel?	They felt no paradox.
How did they get the SinglePairan's mental model?	They could not imagine the DoublePairian's mental model.
Did they themselves use an inconsistent variable symbol?	They did not use inconsistent variable symbol, because they did not understand the expectation formula.
What did let them notice the inconsistent use of a variable symbol?	Analysis of the expectation formula from SinglePairian's point of view, or the thougt that E = (1/2)2x + (1/2)(x/2). is mistake of E = (1/2)2a + (1/2)a. ← Added on July 4,2015.
What did they think the root cause of paradox?	They had not any interest about the root cause. (Revised on October 10, 2015)

Some psychological questions

This paragraph was totally revised on February 7, 2016.

Most important question

On which moment did they feel a paradox?

• At the moment when they read that the expected value of the other envelope is (1/2)2x + (1/2)(x/2) > x.
• At the moment when they read that this calculation does not depend on the value of x, so chosen envelope always has less expected amount of money than another envelope.

Another important question

• When people understand the calculation in the problem how many of them are SinglePairians?
• When Doublepairians read the theory of "E = (1/2)2a + (1/2)a" how many of them switch to SinglePairians?
• Which is the resolution for the people who felt some paradox? ( ← Revised on February 14, 2016)
  Finding mistake of the calculation?
  Guarantee of equivalence of the envelopes?
  Finding correct calculation formula whatever it is?

Another questions

• Let's consider the confusion of the ratio of expected values and expected value of the ratios.
  Isn't the confusion the true cause which made a SinglePairian aware of a paradox? 
  (I explained this question using figures in Appendix : Inconsistent Variable vs Confusion of Expected Ratio.)
• I think that 'Inconsistent Variable' more easily appear in calculation of loss and gain than in calculation of expected value.
  Why does nobody mention this phenomenon ? 
  (Please see Appendix : Inconsistent Variable in Calculation of Loss and Gain.)

A table of the probable pattern of the advocator of the theory of "E = (1/2)2a + (1/2)a"

This paragraph was added on January 2, 2015.

Percentages in the following table mean my expectation.  (These percentages were revised on July 4, 2015, December 13, 201, February 7, 2016, March 2, 2016.)

Property	Patern 1 self-deception 10%	Patern 2 sophism 10%	Patern 3 fictitious paradox 80%
Which phrase had let them feel paradox?	The other envelope is profitable regardless of the amount of money in the chosen envelope.	–	(1/2)(x/2)+(1/2)2x > x
When he/she has understood the expectation formula, which pairian has he/she been?	DoublePairian	– (Revised on April 10, 2016)	–
Has he/she felt some paradox while he/she is a SinglePairian?	–	No	–
Did they understand that the goal is to find any calculation which causes no paradox?	No	No	Yes

Psychological experiment to answer these questions

This paragraph was revised on November 5, 2014, February 7, 2016.

I hope somebody study these questions by psychological experiment.

Experiment using expectation formula

I think that the experiment might have following process.

• Mandatory
  The participants of the experiment read the rule of the game of the two envelopes problem.
• Optional
  The participants read sample of the amounts of money enclosed in each envelopes.
    Example ( $100, $200 )
• Mandatory
  The participants imagine that they select one of the two envelopes.
• Optional
  The participants read sample of specific amount of money enclosed in each envelopes.
    Example ( $200 : the selected envelope, $100 or $400 : the another envelope )
• Optional
  The participants read sample of symbolic amounts of money enveloped in each envelops.
    Example ( $X : the selected envelope, $(X/2) or $2X : the another envelope )
• Optional
  Expression like the following example.
  
  After exchange, your money will be doubled with a probability 1/2 , and it will be halved with a probability 1/2.
  It is mathematically true but may induce inconsistent variable in the brain of the reader.
  ↑ This option was added on March 14, 2015.
• Optional
  The participants consider the amount of money enclosed in the other envelope. Then they write the ratio of the possible largest amount and the possible smallest amount.
  ↑ This option was revised on November 29, 2015.
• Mandatory
  The participants read a expectation formula of the expected value of the amounts of money enclosed in the another envelope which is larger than the amounts of money in the selected envelope.
• Mondatory
  The participants answer whether they understand the calculation.
  ↑ This step was added on February 7, 2016.
• Optional
  The participants answer whether they feel some sense of incongruity.
  ↑ This step was added on February 7, 2016.
• Optional
  The participants answer whether the expetation formula has some mistake.
• Optional
  The participants consider the amount of money enclosed in the other envelope. Then they write the ratio of the possible largest amount and the possible smallest amount.
  ↑ This option was revised on November 29, 2015.
• Mandatory
  The participants read one of following explanations which explain how the expectation formula induce a paradox.
  
  • The theory that inconsistent use of variable symbol is the cause.
  • The theory that wrong assumtion of the probability distribution of amount of money
• Optional
  The participants consider the amount of money enclosed in the other envelope. Then they write the ratio of the possible largest amount and the possible smallest amount.
  ↑ This option was revised on November 29, 2015.
• Mandatory
  The participants point the mistake in the expectation formula.

I hope that the variation of the optional process of the experiment will answer some of my psychological question.

Experiment without expectation formula

I think that 'Inconsistent Variable' more easily appear in calculation of loss and gain than in calculation of expected value.
As follows without expectation formula, we can explain how a paradox occur.


Some participants of such a experiment may point the mistake of such a explanation.
↑ This sentence was revised on March 29,2015.

Four 'Two Envelope Paradoxes'

On March 2 , 2016, title and contents of this paragraph were greatly revised.

The paradoxes of the two envelopes problem might be classified by the following aspects.

• phrase which has let the resolver feel paradox
• whether resolver is SinglePairian

	resolver is SinglePairian	resolver is DoublePairian
The phrase "(1/2)(x/2)+(1/2)2x > x" has let the resolver feel paradox.	paradox by the illusion of objective expectation (fictitious paradox)	paradox by the illusion of objective equivalence (fictitious paradox)
The phrase "The other envelope is profitable regardless of the amount of money in the chosen envelope." has let the resolver feel paradox.	I think that the people who feel this kind of paradox are very few.	standard paradox of the two envelopes problem

In Adition

I have written some pages about associated themes. 
Please see the page 'List of my pages written in English about the two envelopes problem'.

Appendix : Inconsistent Variable vs Confusion of Expected Ratio

We often confuse the ratio of expected values and expected value of the ratios.
I think it is the third cause of the Two Envelope Paradox, and I call it 'Confusion of Expected Ratio'.





I think that we are easy to have 'Confusion of Expected Ratio' rather than 'Inconsistent Variable'.
It is one of the reason why I doubt the Inconsistent variable theory.

Appendix : Inconsistent Variable in Calculation of Loss and Gain

Not using probability, We often calculate loss and gain.

Appendix : LesserOrGreaterian

Some theory says that people may create another mental model as follows,  and that it is the cause of the paradox that such people confuse the mean value of the lesser amount of money and the mean value of the amount in the chosen envelope.

type 1


type 2


type 3
← Added on January 10, 2015.



I think that people who have these mental model should be called "LesserOrGreaterian" or "LesserOrGreaterMeanValuean".

The section "Simple resolutions" in the article "Two envelopes problem" (At the revision 21:39, 23 November 2014) in the English language Wikipedia seemed to say as follows.
↑ Added on January 10, 2015.

• Some people have the mental model of LesserOrGreaterian or LesserOrGreaterMeanValuean.
• They think that the subject matter of the "two envelopes problem" is the magnitude relation of the following values.
  • mean value of the amounts of money in the envelopes which have lesser amount
  • mean value of the amounts of money in the envelopes which have greater amount
  • mean value of the amounts of money in the other envelopes
• And they forgot they were thinking of expectation values under two different conditions as follows.
  
  Let x and y be the amounts in the envelope selected by you and the amount in the another envelope respectively.
  Let X and Y be random variables which take x and y as their value respectively.
  E(Y)=(1/2)E(2X|X<Y) + (1/2)E(X/2|X>Y)
      =(1/2)2E(2X|X<Y) + (1/2)(1/2)E(X|X>Y)
      =(1/2)2E(X) + (1/2)(1/2)E(X)
      =1.25 E(X) > E(X).
  !!! Paradox
  ↑ This table was added on January 24, 2015.

But I am suspicious of such a theory.
I can not imagine that there can be one who can imagine such complicated mental model and make such a confusion.
To my eyes this theory is only distortion.

Some references for the remaining mathematical problems

• Broome,John.(1995).
  The Two-envelope Paradox
  Analysis 55(1): 6–11
• McDonnell, M.D. , Grant, A.J. , Land, I. , Vellambi, B.N. , Abbott, D. And Lever, K.(2011).
  Gain from the two-envelope problem via information asymmetry: on the suboptimality of randomized switching
  Proceedings of the Royal Society
• Norton, J.D.(1998).
  When the sum of our expectations fails us: The exchange paradox.
  Pacific Philosophical Quarterly 79:34–58

Return to the list of my pages written in English about the two envelopes problem

---